A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves

Geophysics ◽  
2007 ◽  
Vol 72 (2) ◽  
pp. T9-T17 ◽  
Author(s):  
Francis H. Drossaert ◽  
Antonios Giannopoulos
Keyword(s):  
Boundary Condition ◽  
Elastic Waves ◽  
Absorbing Boundary Conditions ◽  
Attractive Alternative ◽  
Complex Frequency ◽  
Electromagnetic Modeling ◽  
Absorbing Boundary ◽  
Split Field ◽  
Recursive Integration ◽  
Stretching Function

In finite-difference time-domain (FDTD) modeling of elastic waves, absorbing boundary conditions are used to mitigate undesired reflections that can arise at the model’s truncation boundaries. The perfectly matched layer (PML) is generally considered to be the best available absorbing boundary condition. An important but rarely addressed limitation of current PML implementations is that their performance is severely reduced when waves are incident on the PML interface at near-grazing angles. In addition, very low frequency waves as well as evanescent waves could cause spurious reflections at the PML interface. In electromagnetic modeling, similar problems are circumvented by using a complex frequency-shifted stretching function in the PML formulas. However, in elastic-wave modeling using the conventional PML formulation — based on splitting the velocityand stressfields — it is difficult to adopt a complex frequency--shifted stretching function. We present an alternative implemen-tation of a PML that is based on recursive integration and does not require splitting of the velocity and stress fields. Modeling re-sults show that the performance of our implementation using a standard stretching function is identical to that of the convention-al split-field PML. Then we show that the new PML can be modi-fied easily to include the complex frequency-shifted stretching function. Results of models with an elongated domain show that this modification can substantially improve the performance of the PML boundary condition. An efficient implementation of the new PML requires less memory than the conventional split-field PML, and, therefore, is a very attractive alternative to the con-ventional PML. By adopting the complex frequency-shifted stretching function, the PML can accommodate a wide variety of model problems, and hence it is more generic.

Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T301-T311 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xueyuan Huang ◽  
Yanjie Zhou
Keyword(s):  
Boundary Condition ◽  
Seismic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Absorbing Boundary Conditions ◽  
Second Order ◽  
Complex Frequency ◽  
Absorbing Boundary ◽  
Time Layer ◽  
The Time Domain

The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.

The Split-Field PML Absorbing Boundary Condition for the Unconditionally Stable Node-Based LOD-RPIM Method

2018 ◽  
Vol 17 (10) ◽  
pp. 1920-1924
Author(s):  
Junfeng Wang ◽  
Zhizhang Chen ◽  
Jingcheng Liang ◽  
Yang Wu ◽  
Cheng Peng ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Absorbing Boundary Condition ◽  
Stable Node ◽  
Absorbing Boundary ◽  
Unconditionally Stable ◽  
Split Field

Towards a General-Purpose Open Boundary Condition for Wave Simulations

2011 ◽  
Author(s):  
Bulent Duz ◽  
Rene H. M. Huijsmans ◽  
Peter R. Wellens ◽  
Mart J. A. Borsboom ◽  
Arthur E. P. Veldman
Keyword(s):  
Boundary Condition ◽  
Three Dimensional ◽  
General Purpose ◽  
Absorbing Boundary Conditions ◽  
Design Stage ◽  
Open Boundary ◽  
Computational Domain ◽  
Absorbing Boundary ◽  
Scaled Model ◽  
Artificial Boundaries

For the design of FPSO’s in harsh environments an accurate assessment of the ability of the platform to survive in extreme sea conditions is of prime importance. Next to scaled model tests on the FPSO in waves also CFD capabilities are at the disposal of the designer. However even with the fastest computers available it is still a challenge to use CFD in the design stage because of the large computational resources they require. In that respect to use a small computational domain will improve the turn around time of the computations, however at the expense of various numerical artifacts, like reflection on artificial boundaries in the computational domain. In order to mitigate the reflection properties new absorbing boundary conditions have been developed. The work in this paper is constructed on the previous study about the generating and absorbing boundary condition (GABC) in the ComFLOW project. We present a method to apply the GABC on all the boundaries in a three dimensional domain. The implementation of the GABC in ComFLOW is explained in detail.

Absorbing boundary condition for the elastic wave equation

Geophysics ◽  
1988 ◽  
Vol 53 (5) ◽  
pp. 611-624 ◽  
Author(s):  
C. J. Randall
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Grazing Incidence ◽  
Scalar Fields ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Elastic Wave Equation ◽  
Absorbing Boundary

Extant absorbing boundary conditions for the elastic wave equation are generally effective only for waves nearly normally incident upon the boundary. High reflectivity is exhibited for waves traveling obliquely to the boundary. In this paper, a new and efficient absorbing boundary condition for two‐dimensional and three‐dimensional finite‐difference calculations of elastic wave propagation is presented. Compressional and shear components of the incident vector displacement fields are separated by calculating intermediary scalar potentials, allowing the use of Lindman’s boundary condition for scalar fields, which is highly absorbing for waves incident at any angle. The elastic medium is assumed to be homogeneous in the region immediately adjacent to the boundary. The reflectivity matrix of the resulting absorbing boundary for elastic waves is calculated, including the effects of finite‐difference truncation error. For effectively all angles of incidence, reflectivities are much smaller than those of the commonly employed paraxial absorbing boundaries, and the boundary condition is stable for any physical Poisson’s ratio. The nearly complete absorption predicted by the reflectivity matrix calculations, even at near grazing incidence, is demonstrated in a finite‐difference application.

Absorbing boundary conditions for acoustic media

Geophysics ◽  
1985 ◽  
Vol 50 (6) ◽  
pp. 892-902 ◽  
Author(s):  
R. G. Keys
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Plane Waves ◽  
Basic Element ◽  
Absorbing Boundary Conditions ◽  
Boundary Operator ◽  
Order Differential Operator ◽  
Absorbing Boundary ◽  
Boundary Operators

By decomposing the acoustic wave equation into incoming and outgoing components, an absorbing boundary condition can be derived to eliminate reflections from plane waves according to their direction of propagation. This boundary condition is characterized by a first‐order differential operator. The differential operator, or absorbing boundary operator, is the basic element from which more complicated boundary conditions can be constructed. The absorbing boundary operator can be designed to absorb perfectly plane waves traveling in any two directions. By combining two or more absorption operators, boundary conditions can be created which absorb plane waves propagating in any number of directions. Absorbing boundary operators simplify the task of designing boundary conditions to reduce the detrimental effects of outgoing waves in many wave propagation problems.

An optimal absorbing boundary condition for elastic wave modeling

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 296-301 ◽  
Author(s):  
Chengbin Peng ◽  
M. Nafi Toksöz
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Elastic Wave ◽  
Short Note ◽  
Absorbing Boundary Conditions ◽  
Parallel Computer ◽  
Absorbing Boundary Condition ◽  
Absorbing Boundary

Absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries (Lindman, 1975; Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan et al., 1985; Randall, 1988; Higdon, 1991). We are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage. This is practical for 3-D simulation of elastic wave propagation by a finite‐difference method. Peng and Toksöz (1994) developed a method to design a class of optimal absorbing boundary conditions for a given operator length. In this short note, we give a brief introduction to this technique, and we compare the optimal absorbing boundary conditions against those by Reynolds (1978) and Higdon (1991) using examples of 3-D elastic finite‐difference modeling on an nCUBE-2 parallel computer. In the Appendix, we also give explicit formulas for computing coefficients of the optimal absorbing boundary conditions.

A 3D cylindrical PML/FDTD method for elastic waves in fluid‐filled pressurized boreholes in triaxially stressed formations

Geophysics ◽  
2003 ◽  
Vol 68 (5) ◽  
pp. 1731-1743 ◽  
Author(s):  
Qing Huo Liu ◽  
Bikash K. Sinha
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Time Domain ◽  
Elastic Waves ◽  
Numerical Solutions ◽  
Fdtd Method ◽  
Uniaxial Stress ◽  
Absorbing Boundary Conditions ◽  
Principle Of Superposition ◽  
Absorbing Boundary

A new 3D cylindrical perfectly matched layer (PML) formulation is developed for elastic wave propagation in a pressurized borehole surrounded by a triaxially stressed solid formation. The linear elastic formation is altered by overburden and tectonic stresses that cause significant changes in the wave propagation characteristics in a borehole. The 3D cylindrical problem with both radial and azimuthal heterogeneities is suitable for numerical solutions of the wave equations by finite‐difference time‐domain (FDTD) and pseudospectral time‐domain (PSTD) methods. Compared to the previous 2.5D formulation with other absorbing boundary conditions, this 3D cylindrical PML formulation allows modeling of a borehole‐conformal, full 3D description of borehole elastic waves in a stress‐induced heterogeneous formation. We have developed an FDTD method using this PML as an absorbing boundary condition. In addition to the ability to solve full 3D problems, this method is found to be advantageous over the previously reported 2.5D finite‐difference formulation because a borehole can now be adequately simulated with fewer grid points. Results from the new FDTD technique confirm the principle of superposition of the influence of various stress components on both the borehole monopole and dipole dispersions. In addition, we confirm that the increase in shear‐wave velocity caused by a uniaxial stress applied in the propagation direction is the same as that applied parallel to the radial polarization direction.

A transparent boundary technique for numerical modeling of elastic waves

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 963-966 ◽  
Author(s):  
Jianlin Zhu
Keyword(s):  
Numerical Modeling ◽  
Boundary Conditions ◽  
Elastic Waves ◽  
Wave Equations ◽  
Normal Incidence ◽  
Absorbing Boundary Conditions ◽  
S Wave ◽  
Absorbing Boundary ◽  
S Waves ◽  
Elastic Wave Equations

In numerical modeling of wave motions, strong reflections from artificial model boundaries may contaminate or mask true reflections from the interior model interfaces. Hence, developing a kind of exterior model boundary transparent to the outgoing waves is of critical importance. Among proposed solutions, e.g., Smith (1974), Kausel and Tassoulas (1981), and Higdon (1991), the most widely used may be the Clayton and Engquist (1977) method of absorbing boundary conditions, based on paraxial approximations for acoustic and elastic‐wave equations. However, absorbing boundary conditions make the reflection coefficients zero only for normal incidence, and suppression of reflected S-waves (Clayton and Engquist, 1977) becomes poorer as the ratio of P- to S-wave velocity ([Formula: see text]) becomes larger.

STRESS–VELOCITY COMPLETE RADIATION BOUNDARY CONDITIONS

2013 ◽  
Vol 21 (03) ◽  
pp. 1350003 ◽  
Author(s):  
DANIEL RABINOVICH ◽  
DAN GIVOLI ◽  
THOMAS HAGSTROM ◽  
JACOBO BIELAK
Keyword(s):  
Boundary Condition ◽  
Elastic Waves ◽  
Two Dimensions ◽  
High Order ◽  
Radiation Boundary Condition ◽  
Absorbing Boundary ◽  
Radiation Boundary Conditions ◽  
Long Time ◽  
Radiation Boundary ◽  
The Stability

A new high-order local Absorbing Boundary Condition (ABC) has been recently proposed for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. It is based on the stress–velocity formulation of the elastodynamics problem, and on the general Complete Radiation Boundary Condition (CRBC) approach, originally devised by Hagstrom and Warburton in 2009. The work presented here is a sequel to previous work that concentrated on the stability of the scheme; this is the first known high-order ABC for elastodynamics which is long-time stable. Stability was established both theoretically and numerically. The present paper focuses on the accuracy of the scheme. In particular, two accuracy-related issues are investigated. First, the reflection coefficients associated with the new CRBC for different types of incident and reflected elastic waves are analyzed. Second, various choices of computational parameters for the CRBC, and their effect on the accuracy, are discussed. These choices include the optimal coefficients proposed by Hagstrom and Warburton for the acoustic case, and a simplified formula for these coefficients. A finite difference discretization is employed in space and time. Numerical examples are used to experiment with the scheme and demonstrate the above-mentioned accuracy issues.

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